I suggest that before animals decide to mate, they check the number of animals of the same type nearby. If there are too many of the same type nearby, then they don't mate. Or the probability that they decide to mate is reduced significantly if there are too many nearby.
Rationale: I appreciate that you listened to us and sped up butchering. I really do. However, dealing with domestic animals is still a big chore; you need to memorize and brand them and sort through your list and kill most of them. If you don't keep on top of it, the number of animals can go from a few too many to a ridiculous number in a few days. Then you have a sheep pen that you fear to step foot in, and it takes all day to sort them and cook or store their products. Turning them into soylent green is faster but still takes awhile. Even just killing them and leaving them to rot takes longer than I would like because you can't walk around in a crowded pen; you can't shoo them if you're holding a carcass so you have to hearth out and start over.
I know people will suggest just keeping males and females separate; I do that usually too. But eventually you bring the male back to get another generation, and then if you forget about it for awhile the population explodes. Moving animals between pens can be annoying too.
If you go with this density approach to population control, then people could still have the fast growth we have now if they used larger pens. More casual players could keep smaller pens, have slower quality growth, but have less of an overwhelming chore in keeping animals. Or maybe some other solution would be better, but I would appreciate something to make keeping on top of animal population less onerous. Thanks!